Statistical inference

Statistical inference Definition : generalization from a sample to a population. 2 cases: Is a sample belongs to an hypothetical population? Is two samples belong to the same hypothetical population?

Statistical inference 1st possibility x ? Inference

Statistical inference 2nd possibility x ? Inference

Hypotheses We test H0 H0= Null hypothesis H1 = Alternative hypothesis m = Mean of the population k = Constant 1 H0= Null hypothesis H1 = Alternative hypothesis m1 = Mean of the first population m2 = Mean of the second population 2 We test H0

Decision From the sample(s) we decide if we reject or not the null hypothesis. When we are doing inference we are never certain that we took the right decision Population Sample Decision Identical Different Good Error 2 Error 1

Decision 2 type of errors: 1 – If we inferred that 2 groups belong to two different populations when they don’t. We rejected H0 when H0 was true. 2 – If we inferred that 2 groups belong to the same population when they don’t. We kept H0 when H0 was false. Population Sample Decision Identical Different Good Error 2 Error 1

Sampling Distribution of the Mean 1- Inference about the mean of a population Sampling Distribution of the Mean Sample (n) Sampling Distribution of the Mean Population

Sampling Distribution of the Mean Characteristics: Follows a normal curve. The mean will be equal to the one of the population The standard deviation will be equal to The larger the sample size is, the smaller the standard error will be.

Sampling Distribution of the Mean Sample Sampling Distribution of the Mean Population

Sampling Distribution of the Mean Sample Sampling Distribution of the Mean Population

Sampling Distribution of the Mean Sample Sampling Distribution of the Mean Population

Sampling Distribution of the Mean Sample Sampling Distribution of the Mean Population

Test of Significance If we suppose that the null hypothesis is true, what is the probability of observing the giving sample mean? If it is unlikely, we will reject H0, else we will keep H0. Unlikely: 5% or 1% = a = significance threshold

Test of Significance Example: one side H0: m = 72 H1: m < 72 (based on previous studies) a = 0.05 (5%) s = 9 = 65 n = 36 Because zx is greater za we reject the null hypothesis and accept the alternative hypothesis za = 1.65

Test of Significance Example : 2 sides H0: m = 72 H1: m  72 = 68 n = 36 Because zx is greater za we reject the null hypothesis and accept the alternative hypothesis za = 1.96

Confidence intervals We are never sure that the mean of our sample is exactly the real mean of the population. Therefore, instead of given the mean only, it is possible de quantify our level of certitude by specifying a confidence interval around the mean.

Confidence intervals Example: CI = 95% = 50,7 n = 100 s = 20 Therefore, there is a 95% probability that the mean of the population is between 46.75 and 54.62 za = 1.96

Confidence intervals Example: CI = 99% = 50,7 n = 100 s = 20 Therefore, there is a 99% probability that the mean of the population is between 445.54 and 55.86 za = 2.58

Distribution of sample mean differences 2- Inference for the difference  between two population means distribution of sample mean differences Samples (n) Distribution of sample mean differences Population

Distribution of sample mean differences Characteristics: Follows a normal distribution The mean will be equal to 0 (m1-m2=0) The standard deviation will be equal to:

Decision rule

Test of Significance Example: What is the probability of observed difference between the following groups? H0: m1 = m2 (m1 - m2 = 0) H1: m1  m2 (m1 - m2  0) a = 0.05 (5%) = 50 s1 = 5 n1 = 36 = 48 s2 = 5 n2 = 36

Test of Significance Example: What is the probability of observed difference between the following groups? H0: m1 = m2 (m1 - m2 = 0) H1: m1  m2 (m1 - m2  0) a = 0.05 (5%) = 50 s1 = 5 n1 = 36 = 48 s2 = 5 n2 = 36 Because the observed z is lower than the critical (za) we will keep the null hypothesis

Confidence intervals

Test of Significance Example: a 95% confidence interval H0: m1 = m2 (m1 - m2 = 0) H1: m1  m2 (m1 - m2  0) a = 0.05 (5%) = 50 s1 = 5 n1 = 36 = 48 s2 = 5 n2 = 36

Test of Significance Example: a 95% confidence interval H0: m1 = m2 (m1 - m2 = 0) H1: m1  m2 (m1 - m2  0) a = 0.05 (5%) = 50 s1 = 5 n1 = 36 = 48 s2 = 5 n2 = 36 Therefore there is a 95% probability that the mean difference between the populations is between -0.3128 and 4.3128